UTA-splines and UTADIS-splines Olivier Sobrie 1 , 2 - Nicolas Gillis - - PowerPoint PPT Presentation
UTA-splines and UTADIS-splines Olivier Sobrie 1 , 2 - Nicolas Gillis 2 - Vincent Mousseau 1 - Marc Pirlot 2 1 cole Centrale de Paris - Laboratoire de Gnie Industriel 2 University of Mons - Faculty of engineering July 5, 2016 University of Mons
Olivier Sobrie1,2 - Nicolas Gillis2 - Vincent Mousseau1 - Marc Pirlot2
1École Centrale de Paris - Laboratoire de Génie Industriel 2University of Mons - Faculty of engineering
July 5, 2016
University of Mons
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1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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Additive value function model
1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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Additive value function model
Principle
◮ A score is computed for each alternative ◮ The score is used to rank or to sort alternatives
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Additive value function model
◮ A marginal value function uj is associated to each criterion j ◮ Marginal value functions are monotonic ◮ Marginal value functions are normalized between 0 and 1, s.t.
uj(aj) = 0 and uj(aj) = 1
◮ A weight wj is associated to each criterion j, s.t. n j=1 wj = 1
uj j
+
0aj 1 aj uj(aj) aj
◮ Utility of an alternative a :
U(a) =
n
wj · uj(aj)
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Additive value function model
◮ A marginal value function uj is associated to each criterion j ◮ Marginal value functions are monotonic ◮ Marginal value functions are normalized between 0 and 1, s.t.
u∗
j (aj) = 0 and u∗ j (aj) = wj ◮ A weight wj is associated to each criterion j, s.t. n j=1 wj = 1
u∗
j
j
+
0aj wj aj u∗
j(aj)
aj
◮ We also have :
u∗
j (a) = wj · uj(aj) and u∗ j (aj) = wj ◮ Utility of an alternative a :
U(a) =
n
wj · uj(aj) =
n
u∗
j (aj)
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Additive value function model
600 0.0 0.1 0.2 m u∗
j(aj)
800 0.0 0.1 0.2 m
200 0.0 0.1 0.2 e price 5 45 0.0 0.1 0.2 m2 size 1 5 0.0 0.1 0.2 ⋆ rating
Ranking
Plaza
0.53
≻
Miramar
0.51
≻
Hilton
0.43
≻
Hotel W
0.41
Sorting
Good Bad
Plaza
0.53
Miramar
0.51 0.5
Hilton
0.43
Hotel W
0.41
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Learning an AVF model
1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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Learning an AVF model
◮ UTA : LP for learning the parameters of an AVF-ranking model ◮ UTADIS : LP for learning the parameters of an AVF-sorting model ◮ Other methods : UTA*, ACUTA, . . . ◮ Monotonicity of the marginals is ensured ◮ Marginals are modeled with piecewise linear functions
u∗
j
j
+
0aj
+
u∗1
j
+
g1
j
+
u∗2
j
+
g2
j
+
u∗3
j
+
g3
j
+
wj
+
aj u∗
j(aj)
aj
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UTA(DIS)-poly
1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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UTA(DIS)-poly
Principle
◮ Use of polynomials for the marginal value functions
u∗
j
j
+
0aj
+
u∗1
j
+
g1
j
+
u∗2
j
+
g2
j
+
u∗3
j
+
g3
j
+
wj
+
aj u∗
j(aj)
aj
u∗
j
j
+
0aj wj aj u∗
j(aj)
aj
Motivations
◮ Improve the flexibility of the model ◮ Improve the interpretability of the model
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UTA(DIS)-poly
◮ Use of semi-definite programming (SDP)
◮ Based on interior point methods ◮ Possibility to impose the nonnegativity of a symmetric matrix
Q = q1,1 q1,2 q1,3 · · · q1,n q2,2 q2,3 · · · q2,n q3,3 · · · q3,n ... . . . (symmetric) qn,n ≥ 0
◮ Monotonicity of the marginals guaranteed
◮ Ensured by imposing the nonnegativity of the derivative ◮ Use of Hilbert’s theorems
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UTA(DIS)-poly
Theorem (Hilbert) A polynomial F : Rn → R is nonnegative if it is possible to decompose it as a sum of squares (SOS) : F(z) =
f 2
s (z)
with z ∈ Rn. Theorem (Hilbert) A non-negative polynomial in one variable is always a SOS. Theorem (Hilbert) A polynomial p(x) in one variable x is non-negative in the interval [v1, v2], if and only if p(x) = (x − v1) · q(x) + (v2 − x) · r(x) where q(x) and r(x) are SOS.
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UTA(DIS)-poly
x y a1 10 7 a2 6 8 a3 7 5 a1 ≻ a2 ≻ a3
◮ We define u∗ 1(x) and u∗ 2(y) as third degree polynomials :
u∗
1(x) = px,0 + px,1 · x + px,2 · x2 + px,3 · x3,
u∗
2(y) = py,0 + py,1 · y + py,2 · y2 + py,3 · y3. ◮ Scores of a1, a2 and a3 are given by :
U(a1) = px,0 + 10px,1 + 100px,2 + 1000px,3 + py,0 + 7py,1 + 49py,2 + 343py,3, U(a2) = px,0 + 6px,1 + 36px,2 + 324px,3 + py,0 + 8py,1 + 64py,2 + 512py,3, U(a3) = px,0 + 7px,1 + 49px,2 + 343px,3 + py,0 + 5py,1 + 25py,2 + 125py,3.
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UTA(DIS)-poly
◮ Scores of a1, a2 and a3 are given by :
U(a1) = px,0 + 10px,1 + 100px,2 + 1000px,3 + py,0 + 7py,1 + 49py,2 + 343py,3, U(a2) = px,0 + 6px,1 + 36px,2 + 324px,3 + py,0 + 8py,1 + 64py,2 + 512py,3, U(a3) = px,0 + 7px,1 + 49px,2 + 343px,3 + py,0 + 5py,1 + 25py,2 + 125py,3.
◮ We have a1 ≻ a2 and a2 ≻ a3, which implies :
U(a1) − U(a2) + σ+(a1) − σ−(a1) − σ+(a2) + σ−(a2) > 0, U(a2) − U(a3) + σ+(a2) − σ−(a2) − σ+(a1) + σ−(a1) > 0.
◮ By replacing U(a1), U(a2) and U(a3), we have :
4px,1 + 64px,2 + 776px,3 − py,1 − 15py,2 − 231py,3 + σ+(a1) − σ−(a1) −σ+(a2) + σ−(a2) > 0, −px,1 − 13px,2 − 19px,3 + 3py,1 + 39py,2 + 387py,3 + σ+(a2) − σ−(a2) −σ+(a3) + σ−(a3) > 0.
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UTA(DIS)-poly
◮ We impose the derivative of u∗ 1 and u∗ 2 to be SOS :
u∗′
1 = xTQx
= 1 x T q0,0 q0,1 q1,0 q1,1 1 x
u∗′
2 = yTRy
= r0,0 + (r0,1 + r1,0) y + r1,1y2.
◮ Q and R have to be semi-definite positive, in conjunction with :
px,1 = q0,0, 2px,2 = q0,1 + q1,0, 3px,3 = q1,1, and py,1 = r0,0, 2py,2 = r0,1 + r1,0, 3py,3 = r1,1.
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UTA(DIS)-poly
◮ We add normalization constraints :
px,0 = 0, py,0 = 0, 10px,1 + 100px,2 + 1000px,3 + 10py,1 + 100py,2 + 1000py,3 = 1.
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UTA(DIS)-poly
min σ+(a1) + σ−(a1) + σ+(a2) + σ−(a2) + σ+(a3) + σ−(a3). such that : 4px,1 + 64px,2 + 776px,3 − py,1 − 15py,2 − 231py,3 +σ+(a1) − σ−(a1) − σ+(a2) + σ−(a2) > 0, −px,1 − 13px,2 − 19px,3 + 3py,1 + 39py,2 + 387py,3 +σ+(a2) − σ−(a2) − σ+(a3) + σ−(a3) > 0, px,0 = 0, py,0 = 0, 10px,1 + 100px,2 + 1000px,3 + 10py,1 + 100py,2 + 1000py,3 = 1, px,1 = q0,0, 2px,2 = q0,1 + q1,0, 3px,3 = q1,1, py,1 = r0,0, 2py,2 = r0,1 + r1,0, 3py,3 = r1,1, with :
PSD, σ+(a1), σ−(a1), σ+(a2), σ−(a2), σ+(a3), σ−(a3) ≥ 0.
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UTA(DIS)-splines
1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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UTA(DIS)-splines
Principle
◮ Use of splines for the marginal value functions ◮ Monotonicity of the marginals guaranteed as in UTA(DIS)-poly
u∗
j
j
+
0aj
+
u∗1
j
+
g1
j
+
u∗2
j
+
g2
j
+
u∗3
j
+
g3
j
+
wj
+
aj u∗
j(aj)
aj
u∗
j
j
+
0aj wj aj u∗
j(aj)
aj
+
u∗1
j
+
g1
j
+
u∗2
j
+
g2
j
+
u∗3
j
+
g3
j
Motivations
◮ Improve the flexibility of the model ◮ Generalization of UTA(DIS)-poly and UTA(DIS)
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Experiments
1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research
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Experiments
◮ Implemented in MATLAB ◮ CVX library ◮ SDPT3 solver ◮ Everything is available on https://github.com/oso/uta-matlab
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Experiments
300 600 0.0 0.2 0.4 euro u1 D = 2 300 600 0.0 0.2 0.4 euro D = 6 300 600 0.0 0.2 0.4 euro D = 10 500 1,500 0.0 0.2 0.4 km u2 500 1,500 0.0 0.2 0.4 km 500 1,500 0.0 0.2 0.4 km 50 150 0.0 0.1 0.2 m2 u3 50 150 0.0 0.1 0.2 m2 50 150 0.0 0.1 0.2 m2 real marginal learned marginal
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Experiments
300 600 0.0 0.2 0.4 euro u1 D = 1 300 600 0.0 0.2 0.4 euro D = 2 300 600 0.0 0.2 0.4 euro D = 3 500 1,500 0.0 0.2 0.4 km u2 500 1,500 0.0 0.2 0.4 km 500 1,500 0.0 0.2 0.4 km 50 150 0.0 0.1 0.2 m2 u3 50 150 0.0 0.1 0.2 m2 50 150 0.0 0.1 0.2 m2 real marginal learned marginal
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Experiments
◮ 5 datasets containing 209 to 1728 instances ◮ Instances evaluated on 4 to 10 criteria ◮ Sorting of the alternatives in 4 to 5 categories
Dataset # instances # attributes # categories JRA 172 5 4 CPU 209 6 4 LEV 1000 4 5 SWD 1000 10 4 CEV 1728 6 4
◮ Dataset split 100 times in two disjoint sets (learning set and test set)
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Experiments
Test set (number of pieces - degree - continuity)
Size Dataset 1-1-0 1-2-0 1-3-0 30 % CEV 0.7449 ± 0.0110 0.7662 ± 0.0099 0.7487 ± 0.0109 CPU 0.9195 ± 0.0350 0.9119 ± 0.0319 0.9050 ± 0.0315 JRA 0.5950 ± 0.0537 0.6084 ± 0.0540 0.6248 ± 0.0399 LEV 0.5935 ± 0.0134 0.5902 ± 0.0146 0.5768 ± 0.0164 SWD 0.5408 ± 0.0174 0.5439 ± 0.0201 0.5395 ± 0.0186 50 % CEV 0.7463 ± 0.0115 0.7668 ± 0.0130 0.7502 ± 0.0133 CPU 0.9365 ± 0.0261 0.9275 ± 0.0272 0.9166 ± 0.0279 JRA 0.6134 ± 0.0467 0.6177 ± 0.0487 0.6342 ± 0.0376 LEV 0.5961 ± 0.0145 0.5933 ± 0.0159 0.5810 ± 0.0169 SWD 0.5481 ± 0.0164 0.5495 ± 0.0186 0.5436 ± 0.0195 70 % CEV 0.7444 ± 0.0141 0.7657 ± 0.0136 0.7473 ± 0.0139 CPU 0.9408 ± 0.0235 0.9308 ± 0.0242 0.9234 ± 0.0285 JRA 0.6309 ± 0.0476 0.6315 ± 0.0502 0.6405 ± 0.0414 LEV 0.5982 ± 0.0189 0.5938 ± 0.0188 0.5859 ± 0.0197 SWD 0.5479 ± 0.0183 0.5506 ± 0.0196 0.5470 ± 0.0162
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Experiments
Test set (number of pieces - degree - continuity)
Size Dataset 1-3-2 2-3-2 3-3-2 30 % CEV 0.7487 ± 0.0109 0.7583 ± 0.0100 0.7603 ± 0.0115 CPU 0.9050 ± 0.0315 0.9043 ± 0.0285 0.8988 ± 0.0296 JRA 0.6248 ± 0.0399 0.6236 ± 0.0428 0.6197 ± 0.0478 LEV 0.5768 ± 0.0164 0.5745 ± 0.0184 0.5738 ± 0.0180 SWD 0.5395 ± 0.0186 0.5380 ± 0.0186 0.5380 ± 0.0191 50 % CEV 0.7502 ± 0.0133 0.7607 ± 0.0096 0.7626 ± 0.0105 CPU 0.9166 ± 0.0279 0.9179 ± 0.0290 0.9116 ± 0.0305 JRA 0.6342 ± 0.0376 0.6348 ± 0.0410 0.6385 ± 0.0461 LEV 0.5810 ± 0.0169 0.5780 ± 0.0161 0.5780 ± 0.0166 SWD 0.5436 ± 0.0195 0.5433 ± 0.0184 0.5424 ± 0.0187 70 % CEV 0.7473 ± 0.0139 0.7579 ± 0.0122 0.7597 ± 0.0139 CPU 0.9234 ± 0.0285 0.9262 ± 0.0272 0.9208 ± 0.0273 JRA 0.6405 ± 0.0414 0.6449 ± 0.0429 0.6482 ± 0.0451 LEV 0.5859 ± 0.0197 0.5829 ± 0.0189 0.5815 ± 0.0178 SWD 0.5470 ± 0.0162 0.5451 ± 0.0178 0.5445 ± 0.0171
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Conclusion and further research
◮ Computing time in the same order of magnitude as UTA(DIS) ◮ Generalization of UTA(DIS) ◮ Overfitting effect ◮ Test with other datasets ◮ Use semi-definite programming with other MCDA methods ◮ Source code on https://github.com/oso/uta-matlab
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References
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